The Mathematics of Mozart's Music

Author Mario Livio has studied the relationship between art and mathematics. He tells Michele Norris most of us are attracted to symmetry spiced by some elements of surprise... and that combo is the essence of Mozart's music.

And I'm Michele Norris. You may have noticed that we heard a lot of music composed by Wolfgang Amadeus Mozart during our program. It's our way of celebrating the 250th anniversary of the composer's birth. But what makes Mozart's music so special? Why does great music endure?

Mario Livio has given that some thought. He's an author and an astrophysicist. And when he listens to Mozart, he hears the language of symmetry.

(Soundbite of music)

Mr. MARIO LIVIO (Author and Astrophysicist): So, you notice that, you know, it goes up and then down, and then up again in the same way, and down again in the same way. And then the same thing repeats itself twice. So this is the same type of symmetry you would see in maybe wallpaper design. You know, where you walk a certain distance and you see the same pattern again. Here you wait a little bit, and you hear the same pattern again. Sometimes, you know, it's changed a little bit not to make this boring. So, you know, he changes one note, or he moves a half a note up or down, you know, and so on. In music they call this, transposition. We call this symmetry under translation, which means you move from place to place, you see the same thing.

(Soundbite of music)

NORRIS: Mozart was fascinated with Mathematics. Did that inform his music, or was it actually quite the opposite?

Mr. LIVIO: Not directly. But, you know, his sister, Nanneril, said that when he was a kid at school he could talk about nothing else other than numbers, and he filled the walls of their house with numbers. And in his music, you see, indeed, all these symmetries. You also see, occasionally he writes numbers, you know, at the margin of the pieces of music. There is one piece where he actually calculates the probability of winning the lottery in the margin of the music. But you see the symmetry in his music. It's not just the symmetry that I indicated, the one of like a wall pattern. It's symmetry also of rotation. You know, like a snowflake that you rotate looks the same, sometimes when you look at the score of his music, it looks a bit like, if you take the letter S and turn it on its side.

So you see precisely that symmetry where something is coming down and then going up in precisely the same way.

NORRIS: And so, when you listen to something like Symphony #40, and you sort of pull the lens back and look at the entire composition, you start to see all kinds of symmetry in different forms and different ways.

Mr. LIVIO: That's right. And in many of his pieces, you see that. In fact, in most, I would say, you see this.

(Soundbite of music)

NORRIS: What explains Mozart's total fascination with symmetry?

Mr. LIVIO: We don't know that, of course, but we are aware of the fact that there are some connections between talent in music and in mathematics. For example, there are many prodigies. You know, Mozart was, of course, a prodigy. You know, he composed things at age 8. And he loved mathematics. Similarly, there are many prodigies in mathematics who do their best work, you know, when they are extraordinarily young. It appears that both in music and in mathematics, you know, you don't need to read entire encyclopedias before you become creative. You can, you know, just know a little bit about this, and with the talent that you have, you can immediately generate breakthroughs. And this is what happened with Mozart, or it happened with a number of mathematicians like the French Evariste Galois who, you know, Mozart died at 35, Evariste died at 20. You know, they both did all their best work, you know, when they were teenagers, essentially.

(Soundbite of music)

NORRIS: The symmetry, does that explain why certain kinds of music, certain forms of music are pleasing to our ear?

Mr. LIVIO: It's part of that. It turns out that the music we like best, in all forms of music, not just classical music, is of the type that is kind of in the middle between, you know, being predictable and, and being surprising. I mean, We don't like it to be completely predictable. We definitely don't like to be surprised all the time.

NORRIS: And the surprise in his music, in Mozart's music, in fact, writers often refer to it as the Mozart effect, doing exactly the opposite of what you expect the artist to do.

Mr. LIVIO: Right. So, right, so that's the element of surprise, indeed. I must say, I find myself that as I grow older, I like things to be a little bit more predictable than I used to when I was younger.

NORRIS: I wonder if artists instinctively try to achieve this. It's almost like they're drawn to it; it's almost then within our DNA, our artistic DNA.

Mr. LIVIO: It is. But artists also know that we have this. So occasionally, for example, especially in 20th century art, when they actually want indeed to surprise us or to shock us, they especially, you know, get away from symmetry. You know, if I wear my glasses in a way that they are completely tilted, you will really have a hard time sitting with me for a long time. You will find this very, very disturbing. And artists sometimes use that, both in music and in other arts, you know, where they deviate from the symmetry just to shock us.

NORRIS: You think about the sort of canon of Mozart music. What would be the most complex example of symmetry within the sort of the body of the composition?

Mr. LIVIO: Just in terms of pure mathematics, perhaps his most surprising piece is the one that's called the Musical Dice Game. It's a minuet with 16 measures. He fixed the last measure, and for one measure he gave two possibilities of how to play it. But then he actually wrote 11 other different measures, and you're supposed to roll two dice to decide which one to choose to play. Now, if you roll two dice, you know, and you get the number, you get two and three, so you take number five. So now you play number five. Then you roll the dice again, it turns out to be eight. So now you play, in the next measure you play number eight, you know, and so on. There are literally hundreds of trillions of possibilities to choose from to play this.

So, in fact, if you'd like, I mean, nobody has yet heard all the possibilities of this particular minuet. You know, but he wrote it in such a way that no matter which one you choose by rolling the dice, it still sounds very, very nice.

NORRIS: What we're talking about though is random selection, not symmetry.

Mr. LIVIO: Right, right. But mathematics, still mathematics.

NORRIS: Well, Mario, it's been wonderful talking to you. Thanks so much for coming in to speak to us.

Mr. LIVIO: Thank you very much, Michele.

NORRIS: Mario Livio's latest book is called The Equation that Couldn't be Solved: How A Mathematical Genius Discovered the Language of Symmetry.

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